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MOX-Report No. 31/2015
Domain-selective functional ANOVA for processanalysis via signal data: the remote monitoring in laser
welding
Pini, A.; Vantini, S.; Colombo, D.; Colosimo, B. M.; Previtali,
B.
MOX, Dipartimento di Matematica Politecnico di Milano, Via Bonardi 9 - 20133 Milano (Italy)
[email protected] http://mox.polimi.it
Domain-Selective Functional ANOVA for Process
Analysis via Signal Data: the Remote Monitoring in
Laser Welding
A. Pinia, S. Vantinia, D. Colombob, B. M. Colosimob, B. Previtalib
a MOX– Department of Mathematics, Politecnico di Milano, Milan Italy
b Department of Mechanical Engineering, Politecnico di Milano, Milan Italy
Abstract
In many application domains, process monitoring and process optimiza-tion have to deal with functional responses, also known as profile data. Inthese scenarios, a relevant industrial problem consists in discovering whichspecific parts of the functional response is mostly affected by the processchanges. As a matter of fact, knowledge of the specific locations where thecurve is more sensitive to process changes can bring several advantages.It can be exploited to design specific monitoring devices directly focusingon the functional data pertaining to the selected intervals. Secondly, thedimensional reduction can eventually bring to an increase of the power todetect process changes.
This paper proposes a methodology to inferentially select the parts ofthe output functions that are more informative in terms of the underly-ing factors. The procedure is based on a non-parametric domain-selectiveANOVA for functional data, which results in the selection of the intervalsof the domain presenting statistically significant effects of each factor. Toillustrate its potential in industrial applications, the proposed procedureis applied to a case study on remote laser welding, where the main aim ismonitoring the gap between the welded plates through the observation ofthe emission spectra of the welded material.Keywords: Statistical Process Control, Design of Experiments, FunctionalData Analysis, Family-Wise Error Rate
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1 Introduction
Due to the advent of in-line sensoring and non-contact measurement systems,more and more often responses of manufacturing processes can be modeled asfunctional data, i.e., a vector of data, where each value corresponds to a specifictemporal or spatial location. Functional data can in fact represent the shape ofa machined feature and/or the pattern of a signal acquired by the process overtime. In this scenario, statistical quality monitoring and process optimizationhave to be appropriately rethought to deal with functional responses. Withreference to statistical quality monitoring, the problem of checking the stabilityof a functional response has been named profile monitoring and has attracted theattention of many researchers over the past ten years (for an overview, see, forinstance, Noorossana et al. 2012 and references therein). In the field of processoptimization, robust optimization of functional data received the attention ofresearchers in many different application domains (Nair et al. 2002; Del Castilloand Colosimo 2011; Alshraideh and Del Castillo 2014; Del Castillo et al. 2012;He et al. 2014; Drignei 2010).
Both process monitoring and optimization of functional responses share acommon basic problem, i.e., to detect a change of the functional response pat-tern. In the first case, the change is deemed to be due to an out-of-controlstate of the manufacturing process. In the case of process optimization, thefunctional response change is pursued by changing the process parameters, withthe aim of making the response following as close as possible a target pattern.Within such framework, the analysis of the whole response function can providelots of information on the underlying interest factors. Nevertheless, the func-tional observation and analysis present some issues. For instance, the in-processmonitoring of the whole output function can be computationally not affordable,and/or the effect of the factors on the whole function can be difficult to study.Hence, it is often required to reduce the dimensionality of the output data,to provide a methodology that is efficient and actually applicable in industrialenvironments.
Functional principal component analysis (Ramsay and Silverman 2005) canbe a way to reduce the dimensionality, and to select the most informative featuresof the data in terms of the proportion of explained variance (i.e., the functionalprincipal components). This method can be applied to problems dealing withprofile data, to reduce the dimensionality of the output functions and betterunderstand their characteristics and evolution during the industrial process (Hallet al. 2001; Colosimo and Pacella 2007, 2010; Yu et al. 2012; Grasso et al. 2014).Unfortunately, functional principal components are often difficult to interpret,and in the field of functional ANOVA, this analysis can be misleading, as thecomponents of the signal explaining the higher variability are not necessarily theones that present more information about the factors of interest of the industrialprocess. Furthermore, functional principal component analysis relies on theevaluation of all of the original functional data. This means that the entire data
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domain has to be acquired to detect a process change.In many industrial applications, the output functions may be influenced by
the factors only in some parts of their domain, or in different ways over the do-main. To efficiently monitor the process, it is then key to select the informativeparts of the functions (i.e. the ones influenced by the factors). The informationon which parts of the domain are informative can be used to design specificmonitoring devices that only focus on the data pertaining to those parts. Thiswould provide a direct gain, both in terms of efficiency of the monitoring pro-cedure, and in economic terms, since the monitoring would then be based onthe univariate output of the sensor, instead of on more complicated evaluationsbased on the whole functions. As an example, the motivating case study of thisresearch paper is remote laser welding, where the laser emission spectrum is usedas reference to check whether the distance between the plates to be welded (i.e.,the gap) is appropriate. In this case, a previous study Colombo et al. (2013)showed that the gap between the plates affects the emission spectra and hencethis last information can be used for monitoring purposes. However, developinga specific sensor aimed at detecting the emission just at some specific wavelengthinstead of using the whole spectra can reduce costs of the monitoring device andreduce the computational time.
Even though the selection of the informative parts of the domain can bedone in naive ways, for instance by a visual inspection of the data, or based onprior knowledge about the process, in most cases visual inspection and/or priorknowledge can be unclear or misleading, and may neglect important aspects.Hence, we here propose a statistical technique to approach the domain selection.The approach that we propose is sound even in absence of prior knowledgeabout the process, and gives clear and reliable results. We propose to selectthe informative parts of the functions by means of the inferential interval-wisetesting procedure, first proposed in Pini and Vantini (2015), and here extendedto the case of multi-way functional ANOVA. In detail, we test the significance ofthe effects of the factors on the output functions. More importantly, we providea selection of the parts of the domain presenting significant effects of each factor.Once this selection has been done, we propose to monitor the industrial processbased on the information that the functions contain on the selected domains.
The paper is structured as follows: Section 2 presents the statistical method-ology applied for the domain-selective functional ANOVA. Section 3 reports anapplication of the proposed method on a remote laser welding data set. In de-tail, Subsection 3.1 reports the description of the industrial application and therelated problems, Subsection 3.2 reports the description of the data, and Subsec-tion 3.3 reports the results of the domain-selective functional ANOVA on thesedata. A robustness analysis is reported in the Appendix.
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2 Domain-selective functional ANOVA
In this section we describe the methodology that we apply to test a multi-wayfunctional ANOVA. For ease of notation, and without loss of generality, weconsider the example of a two-way functional ANOVA with interaction. Nev-ertheless, the methodology presented here can be directly generalized to theanalysis of factorial design with more than two factors.
Consider a two-way ANOVA in which we investigate the effects of factors Aand B (with I and J levels, respectively), on a functional response, based on areplicated full factorial design. We assume that the response of the model is afunction of a continuous variable t observed over a domain (a, b) ⊂ R. In theconsidered model, the functional response will be expressed as the result of twomain effects, and of an interaction between the previous ones. The aim of theanalysis is to test the significance of every term in the model (i.e., interactionand main effects).
In detail, let yijl : (a, b) 7→ R be the output functional data, where i denotesthe level of the first factor, j denotes the level of the second factor, and l thereplicate. We here assume functional data to be continuous functions of thevariable t. Note that, as shown in Pini and Vantini (2015), this condition canbe relaxed, and the methodology can be applied to generic L2 functions. Thefunctional ANOVA model that we want to test is thus the following:
yijl(t) = µ(t) + αi(t) + βj(t) + γij(t) + εijl(t), (1)
where t ∈ (a, b), with i = 1, . . . , I, j = 1, . . . , J , l = 1, . . . , nij , nij being thenumber of replicates for ith level of the first factor and jth level of the secondfactor. In model (1), µ(t) is the functional grand mean, αi(t) and βj(t) arefunctional main effects, and γij(t) is the functional interaction effect. The func-tional errors εijl(t) are assumed to be independent and identically distributedzero-mean random functions. Note that we do not require the errors to followa gaussian process. All effects (as well as the errors) are expressed as functionsof the continuous variable t. For sake of identifiability, we require the classicalconstraints on the effects, i.e., ∀t ∈ (a, b):
∑Ii=1 niαi(t) = 0;
∑Jj=1 njβj(t) = 0;∑I
i=1
∑Jj=1 nijγij(t) = 0, where ni =
∑Jj=1 nij denotes the number of units at
ith level of the first factor and nj =∑I
i=1 nij denotes the number of units atjth level of the second factor.
The aim of our analysis is to test the significance of all coefficients of model(1). In particular, we want to perform the functional counterparts of uni/multivariateANOVA tests:
• a functional test of the null model, jointly for all factors:{H0,Model : αi(t) = βj(t) = γij(t) = 0 ∀i = 1, . . . , I; j = 1, . . . , J ; t ∈ (a, b);
H1,Model : (H0,Model)C ;
(2)
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• three functional tests for the effects of each factor and interaction:
H0,A : αi(t) = 0 ∀i = 1, . . . , I; t ∈ (a, b); H1,A : (H0,A)C
(3)
H0,B : βj(t) = 0 ∀j = 1, . . . , J ; t ∈ (a, b); H1,B : (H0,B)C
(4)
H0,AB : γij(t) = 0 ∀i = 1, . . . , I; j = 1, . . . , J ; t ∈ (a, b); H1,AB : (H0,AB)C .(5)
Note that, similarly to uni/multivariate two-way ANOVA, (2) is a test of signif-icance of the whole functional model, whereas the three tests (3-5) allow to per-form a model selection. The main difference with respect to a uni/multivariateANOVA is that, being the response functional, tests (2-5) involve functionalcoefficients, i.e., the null hypothesis is rejected whenever there is a significantdifference between the corresponding groups in at least one interval of the do-main.
The problem of testing a functional ANOVA model has been widely dis-cussed in the literature of the last decades, and it can be addressed in sev-eral ways (e.g.,Cuevas et al. (2004); Cuesta-Albertos and Febrero-Bande (2010);Abramovich and Angelini (2006); Antoniadis and Sapatinas (2007)). A com-mon feature of all these works, is that the final result from the ANOVA testingdetermines whether the hypotheses (2-5) are globally accepted or rejected. Inparticular, by applying these tests, we only are able to answer the question “Arethere any statistically significant effects of factors A and/or B on the functionalresponses?”. In the case of a positive answer, these tests are not able to selectthe intervals of the domain in which the effects are detected.
On the contrary, the principal aim of this paper is to provide practitionerswith a methodology that selects the most informative parts of the functionsdomain, in order to use this information to design specific monitoring devices.In detail, for each tests (2-5), in case of rejection of the null hypothesis, we wantto select the intervals of the variable t where significant differences are detected.For this reason, we extend the interval-wise testing proposed in (Pini and Vantini2015), that is a testing procedure for functional data that enables to select theintervals of the domain presenting significant effects.
Another advantage of the application of such procedure is being a non-parametric procedure. In particular, we neither need to assume the normalityof the residuals of the model, nor to specify their covariance structure, that canbe both difficult to assess in the practice.
The extension of the procedure on functional ANOVA is based on three steps,detailed in the following paragraphs.
First step: interval-wise testing. A functional test corresponding to tests(2-5) is performed on any interval of the domain. In detail, given any I ∈ (a, b),
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we perform a test of the null model, and three tests on the main effects andinteraction:{
HI0,Model : αi(t) = βj(t) = γij(t) = 0 ∀i = 1, . . . , I; j = 1, . . . , J ; t ∈ I;
HI1,Model : (HI0,Model)C
(6)
HI0,A : αi(t) = 0 ∀i = 1, . . . , I; t ∈ I; HI1,A : (HI0,A)C (7)
HI0,B : βj(t) = 0 ∀j = 1, . . . , J ; t ∈ I; H1,B : (HI0,B)C (8)
HI0,AB : γij(t) = 0 ∀i = 1, . . . , I; j = 1, . . . , J ; t ∈ I; HI1,AB : (HI0,AB)C . (9)
As proposed in Pini and Vantini (2015), we perform each test in a permuta-tion framework. In detail, we apply permutations of the residuals of the reducedmodel, according to the Freedman and Lane permutation scheme (Freedmanand Lane 1983). As test statistics, we compute the integral over the interval Iof the two-way ANOVA statistics of the corresponding classical F -tests. Thisprovides exact tests for H0,Model, and approximated (asymptotically exact) testsfor H0,A, H0,B and H0,AB. In the following, we denote with pIModel, p
IA, pIB, and
pIAB the p-values of tests (6-9), respectively.
Second step: computation of the adjusted p-value function. An ad-justed p-value function is computed for each test (2-5). In detail, for eacht ∈ (a, b), the p-value p̃(t) is computed as the maximum p-value of all interval-wise tests (6-9) on intervals containing t:
p̃Model(t) = supI3t
pIModel, p̃A(t) = supI3t
pIA, p̃B(t) = supI3t
pIB, p̃AB(t) = supI3t
pIAB.
The adjusted p-value function p̃Model(t) is provided by a control of theinterval-wise error rate, while the adjusted p-value functions p̃A(t), p̃B(t), andp̃AB(t) are provided with an asymptotic control of the interval-wise error rate,as defined in Pini and Vantini (2015).
Third step: domain selection. The intervals of the domain presenting arejection of the null hypothesis are obtained by thresholding the correspond-ing adjusted p-value function at level α: we select intervals presenting at leastone significant effect by thresholding p̃Model(t), intervals presenting a significanteffect of the A factor by thresholding p̃A(t), and so on.
The control of the interval-wise error rate allows controlling the probabilityof detecting false positive intervals, i.e., the probability of wrongly rejecting anyinterval. For instance, since p̃Model(t) is provided by a control of the interval-wise error rate, by selecting the intervals associated with an adjusted p-valuep̃Model(t) ≤ α, we have that, given any interval in which the response is notinfluenced by any factor, the probability that this interval is (wrongly) selectedas significant is lower than α.
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3 Case study: remote monitoring of remote laserwelding
To illustrate the potential of the approach proposed in this paper in industrialapplications, in the following we report an application on remote laser weldingdata. During the remote welding of zinc-coated steel in the lap-joint configura-tion, we register as output profile the optical emission of the welded material asa function of the wavelength. We are interested in selecting the informative partof these profiles for the monitoring of the gap between the two welded surfaces,by taking into account the location in which the emission spectra are acquired.In the following, we report the description of the industrial process of remotelaser welding, and the related issues (Subsection 3.1), the description of the data(Subsection 3.2), and the results of the analysis (Subsection 3.3). A robustnessstudy with respect to the smoothing parameter is reported in the Appendix.
3.1 Remote monitoring of laser welding
Laser welding technologies are quickly replacing conventional welding processes.Furthermore, nowadays the laser welding is often used in remote configurations,i.e., configurations in which the laser beam is moved along the seam with thehelp of a laser scanner. One of the most common applications of such processis the welding of zinc-coated steel in the lap-joint configuration. However, thisparticular configuration and materials present a lot of technical issues, sincethe boiling temperature of the zinc is significantly lower than the one of steel(approx. 906◦C and 1500◦C, respectively). Consequently, during the welding,highly pressurized zinc vapors are often produced at the interface of the twometal sheets, and may cause defects in the welded material, such as spattersand porosities, that can compromise its quality.
The classical solution applied in industrial processes to prevent these defectsis to leave a small gap (order of hundreds microns) between the two metal sheets,to facilitate the degassing (Akhter et al. 1991; Steen et al. 2003). One of themethods used to produce such gap is laser dimpling, which uses a remote pulsedlaser to generate protuberances on one of the plates (Daimler Chrysler AG 2005;Schwoerer 2008; Gu 2010). However, the variance of the height of laser-dimplescan cause errors in the final gap dimension, which can cause defects on the weldedmaterial, compromising its quality and causing variations in the mechanicalproperties of the weld bead.
In Colombo et al. (2013), a method to remotely monitor the gap in remotelaser welding, avoiding destructive off-line tests, is proposed. According to thistechnique, optical emissions are monitored during remote laser welding by aspectroscope. Then, from the acquired spectra, different indicators, or summa-rizing variables, are evaluated, such as the overall emission across the consideredrange, and the emissions in separated wavelength ranges (defined by physicalevaluations of the welding process). For each of the obtained variables, univari-
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ate analysis of variance is performed, and the statistical significance of the effectsof the gap value is used to compare the tested methods. The optical emissionrecorded during the experiment can also depend on the location in the weldseam where the spectrum is acquired. That is why, in Colombo et al. (2013),a two-way analysis of variance is performed, to evaluate the effect of the gapon the emission taking into account the different locations. The former analysisis a valid instrument to assess how the emission is influenced by the gap, andthe results can be used to provide an indicator to evaluate the gap remotely.However, the choice of the better indicator to evaluate the gap effect is difficultto perform, since the analyzed wavelength bands are priorly fixed.
Here, we perform a domain selective functional ANOVA, according to theprocedure described in Section 2, to select the wavelength bands presentingsignificant effects of the gap and the location on the emission, and controlling theprobability of false discoveries. The direct result of this analysis is the selectionof the wavelength bands that are informative in terms of the gap between theplates. The selected bands can thus be used to remotely monitor the gap betweenthe plates during the welding process.
3.2 Experimental procedure and data acquisition
A Through Optical Combiner Monitoring architecture (Capello et al. 2008;Colombo and Previtali 2009, 2010) is used to monitor the laser welding. Ac-cording to this technique, the monitoring of laser emission is performed remotely.Indeed, far from the work area, the optical emissions from the welding processare directly observed inside the laser source through the optical combiner of thefiber laser source with a spectroscope. An extensive description of the experi-mental welding procedure and the monitoring technology goes beyond the scopeof this paper, and can be found in Colombo et al. (2013).
The main objective of this study is to assess the effects of both gap andlocation on the emission functions. To analyze these effects, the emission isacquired in correspondence of different levels of gap and location, in a repeatedfactorial design. Three values of gap, corresponding to 100 nm, 200 nm and300 nm are explored. For each of the analyzed gap values, three replicatesare produced, for a total of nine welded specimens. Inside each specimen, fiveemission spectra are acquired at five different locations, for a total number of 45acquired spectra. The emission data are described in detail in Colombo et al.(2013).
To record the optical emission in the visible range, optical emission spec-troscopy, i.e., the analysis of emitted light with high-wavelength resolution, isused. The laser emission yijl(t) is acquired at 703 discrete wavelengths t between400.521 nm and 800.030 nm (where indexes i, j, l indicate the levels of gap, loca-tion and replicate, respectively). The acquired data, as well as the three means,corresponding to the three different values of the gap, are represented in Fig-ure 1. In the figure, the curves of each color correspond to the 15 functional
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Em
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.)
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Figure 1: Emission data captured in the factorial laser welding experiment (dashed lines),colored according to the three different gaps; and means of the three groups, corresponding tothe three different gaps (solid lines). The figure is divided along the abscissa into the plasma(light blue), laser (light green), and thermal (light red) emissions.
emissions (five locations and three replicates) within each gap level.Note that in the explored wavelength range, it is possible to distinguish
between three different emission ranges:
• between 400 nm and 530 nm, it is observed the emission related to elec-tronic transition, i.e., the plasma emission (light blue area of Figure 1);
• around 535 nm we observe a strong emission line, corresponding to thelaser emission (light green area of Figure 1);
• above 540 nm we observe the emission due to the thermal black-bodyradiation, i.e. the thermal emission (light red area of Figure 1).
The aim of the following analysis is to assess whether the gap and the locationhave some effects on the emission, by taking into account the whole functionalshape of the spectrum, controlling the probability of false discoveries. Finally,we want to locate possible wavelength bands in which the significant effects aredetected.
3.3 Results of the tests
In the application, the emission functions yijl(t) (i = 1, . . . , 3, j = 1, . . . , 5,l = 1, . . . , 3) are modeled according to model (1), where t is the wavelength,αi(t) is the gap functional effect, βj(t) is the location functional effect, γij(t) isthe interaction functional effect, and εijl(t) is the functional error term.
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Figure 2: Left: adjusted p-value function for the tests on H̃0,GapLoc. Right: emission datacolored according to gap and location levels.
We apply the domain-selective functional two-way ANOVA described in Sec-tion 2 to test of the null model H0,Model (2), the gap effect H0,Gap = H0,A (3), thelocation effect H0,Loc = H0,B (4), and the interaction effect H0,GapLoc = H0,AB
(5). The functional data that we analyze are based on a smoothing of the 703discrete data points on a dense piece-wise linear B-spline basis characterized by200 knots. The obtained data after the smoothing are reported in the rightpanels of Figure 3. A robustness analysis with respect to the number of knotsof the expansion is reported in the Appendix. The test statistics of tests onintervals were approximated through a rectangle integration method. Thanks tothe continuity of test statistics with respect to the extremes of integration, theadjusted p-value functions were approximated through a finite family of testsover a fine discrete grid of 703 points (i.e., the maximal resolution of the mon-itoring instrument). The permutation p-values were estimated by means of aConditional Monte Carlo algorithm based on 1000 iterations.
We started by applying the testing methodology described in Section 2 totest the interaction term γij(t). The results of the test are reported in Figure2. In detail, the left panel reports the adjusted p-value function for the testof H0,GapLoc, and the right panel reports the emission data obtained after theB-spline smoothing. Since the interaction effect is not significant (i.e., the ad-justed p-value function presents high values along all the wavelength domain),we decided removing this term from the model, and tested the following additivefunctional ANOVA model:
yijl(t) = µ(t) + αi(t) + βj(t) + εijl(t), ∀t ∈ (a, b). (10)
In the following, we describe in detail the results of the tests on the additivemodel (10). Note that, in this case, the test of the null model becomes:{H̃0,Model : αi(t) = βj(t) = 0, ∀i = 1, . . . , 3; j = 1, . . . , 5; t ∈ (400.521, 800.030);
H̃1,Model : (H̃0,Model)C .
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Figure 3 reports the results of the additive two-way functional ANOVA ofemission data. In particular, the left panels report the adjusted p-value functionsfor each test. In detail, on the top panel, we report the adjusted p-value func-tions of H̃0,Model, on the middle panel the adjusted p-value functions of H0,Gap
and on the bottom panel the adjusted p-value functions of H0,Loc. For ease ofvisualization of the test results, the right panels of the figure report the emissiondata and the significant intervals detected at 5% and 1% levels (areas coloredin light and dark grey, respectively) for the three tests. Data are colored in thethree panels according to the corresponding tests: on the top panel data arecolored differently according to the different levels of both gap and location; onthe middle panel data are colored according to the different levels of gap; on thebottom panel data are colored according to the different levels of location.
Focusing on the test of H̃0,Glob (i.e., top panels of Figure 3), we find a sig-nificant effect of at least one factor among gap and location in nearly the entirewavelength domain. This result suggests that, as expected, the welding con-ditions have a significant effect on the spectrum for all three emission ranges(plasma, laser, and thermal). The gap has a significant effect in most of thewavelength domain (i.e., test of H0,Gap, middle panels of Figure 3), suggestingthat, consistently with previous results, the emission is significantly influencedby the gap on all three emission ranges. The location has a significant effectmostly in the plasma emission range (i.e., test of H0,Loc, bottom panels of Fig-ure 3). This suggests that plasma emission is influenced by the location.
The procedure described in this paper improves the results found by Colomboet al. (2013), by precisely locating the wavelength bands associated to significanteffects of the two factors. Indeed, the most important result highlighted by thisanalysis, and completely new with respect to the literature in this field, is that,looking at all tests together, we detect a band (i.e., the band t ∈ (547 nm,681 nm) corresponding approximately to the thermal emission), in which thegap effect is significant and the location one is not. This suggests the use ofemission data on this band to monitor the gap between the plates during thewelding process at any possible location.
Note that, for instance, the adjusted p-value function of the test of H0,Glob islower than 5% on the interval (547 nm, 681 nm). Thanks to the control of theinterval-wise error rate, if the emission were effected by neither the gap nor thelocation in such band, we would have selected it as significant with a probabilitylower that 5%. Furthermore, since the adjusted p-value function of the testof H0,Gap is lower than 1% on the interval (547 nm, 681 nm), thanks to theasymptotic control of the interval-wise error rate, if the gap were not affect theemission in such band, we would have selected it as significant with a probabilityapproximately lower that 1%.
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Figure 3: Left: adjusted p-value functions for the tests on H̃0,Model (top), H0,Gap (middle) andH0,Loc (bottom). Right: emission data colored according to gap and location levels (top), gaplevels (middle) and location levels (bottom). The gray areas represent significant intervals at5% and 1% levels (light and dark grey, respectively)
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4 Conclusions
We proposed a methodology that can be applied in profile monitoring and func-tional response optimization to select the informative parts of the output func-tions. The methodology exploits a domain-selective functional ANOVA to studythe effects of one or more underlying factors on the output functions. This se-lection is performed by extending the interval-wise testing procedure (Pini andVantini 2015) to the case of multi-way ANOVA framework. This methodologyrepresents a significant improvement with respect to the state of the art tech-niques. Indeed, it is a functional technique, in the sense that considers the wholefunctional data instead of performing a prior dimensional reduction. Further-more, it selects the informative parts of the output functional data, in orderto optimize the monitoring of the input factors through the observation of theoutput curves.
The proposed methodology is applied to a remote laser welding case study.The aim of this study is to remotely monitor the gap between two weldedplates, by observing the emission spectra of the welded material. We appliedthe domain-selective functional two-way ANOVA to study the effects of the gapon the output functions, by taking into account the different locations at whichthe emission is registered.
The final result of this analysis is the selection of a wavelength band (i.e., theband t ∈ (547 nm, 681 nm) corresponding to the thermal emission), in whichthe emission is significantly influenced by the gap and not by the location. Thissuggests to directly record the emission data on this band, to monitor the gapbetween the plates during the welding process at any possible location. Theinterval-wise control, that this procedure is based on, gives a direct measure ofthe reliability of this result. In detail, we know that, if the emission were effectedby neither the gap nor the location in the band t ∈ (547 nm, 681 nm), we wouldhave selected it as significant with a probability lower that 5%.
The robustness of this result with respect to the number of knots p of the B-spline basis expansion employed to smooth the data is discussed in the Appendix.Even though the results of this robustness study show that in this case, thenumber of knots is not a crucial parameter, it is of course of major interest forfuture research to explore the trade-off, in terms of power of the procedure, asp increases.
An interesting future challenge posed by this data analysis is to introducethe gap in the functional model as a numeric explanatory variable, instead of afactor. Since the effect of the gap on the emission spectra is observed to be notmonotonic, to perform this analysis, it is key to study its effect, and add the gapin the model in a non-linear way.
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Appendix. Robustness analysis
To investigate the robustness of the results with respect to the number of knotsof the B-spline expansion, we performed the test by varying this parameter. Weconsidered different cases, based on B-splines expansions with a different numberof knots. In particular, we started from the analysis reported in Section 3 basedon 200 knots, and tried to double or halve the number of knots. Finally, we alsotried the maximum possible resolution (i.e., 703 knots).
The test on the interaction H0,GapLoc is not significant in all cases. Hence,the comparison is made on the additive model without interaction (10). Theresults of the tests are reported in Figure 4. Similarly to the right panels ofFigure 3, the results of the tests of H̃0,Model, H0,Gap and H0,Loc at 1% and 5%levels are represented by means of the gray bands below each graphic. The axison the left indicates the number of knots used for the B-spline basis expansion,i.e., 100, 200, 400, and 703 knots, corresponding to length of B-spline supportsof 8, 4, 2, and 1.14 nm.
This further analysis shows that the results are robust with respect to thenumber of knots. Both the test on the null model H̃0,Model and the test onthe gap H0,Gap report significant differences in most of the wavelength domain.In particular, the gap has a significant effect on all three emission ranges. Onthe other hand, the effect of the location is significant mostly on the plasmaemission. In all cases, the thermal band detected in Subsection 3.3, presentinga significant effect of the gap but not of the location, is preserved.
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